Arbitrary function generators



Feb. 3, 1959 n. L. HERR ARBITRARY FUNCTION GENERATORS 2 Sheets-Sheet 1Filed June 15, 1955 /NVEN7'O,Q DONALD L. HERR Feb. 3, 1959 D. L. HERR2,872,602

ARBITRARY FUNCTION GENERATORS Filed June 13, 1955 2 Sheets-Sheet 2 400CONDUCTORS INVEN TOR. DONALD L HERR BY W a pn/fiuaz ATTORNEYS UnitedStates Patent ARBITRARY FUNCIIGN GENERATORS Donald L. Herr, SantaMonica, Calif. Application June 13, 1955, Serial No. 514,882

12 Claims. (Cl. 310-111) This invention relates to electromagneticinduction transfer devices and more particularly, to novel transferdevices capable of synthesizing arbitrarily chosen continuousmathematical functions.

Electromagnetic voltage and torque transfer devices for generatingcertain specific elementary functions are well known in the art and findwide application in analog computers and fire control systems.Basically, these devices comprise. a rotor body secured to a controlshaft and concentrically positioned for rotation Within a surroundingstator body- Both the. rotor and. stator are provided with discretelydistributed conductor windings. Bythis: arrangement, the degree ofinductive coupling between the rotor and stator windings, when one orthe other is electrically excited, is dependent upon the angularrelationship. of the rotor with respect to the stator.

A typical electromagnetic transfer device of the abovev type now ingeneral use, is the electromagnetic resolver described and claimed in myU. S. Patent No. 2,608,682, entitled Electromagnetic Resolver, issuedAugust 26, 1952. In the resolver, the rotor and stator are each providedwith two windings mechanically 90' degrees apart. An alternating inputsignal applied to. one of the stator windings will result in an outputvoltage from one of the rotor windings proportional to the product ofthe initial input signal and the sine or cosine of the angle of therotor shaft or windings with respect to the stator. Theer is thusprovided a device which may be used to continuously compute the sine orcosine of a varying angle.

The present invention has as its primary object the provision of anelectromagnetic transfer device capable of generating any desiredarbitrary function of the rotor shaft angle, providing such function iscontinuous, thereby resulting in a vastly more universal instrument thanhas heretofore been available.

More particularly, an object of the invention is to provide anelectromagnetic transfer device which may be pro-designedv in accordancewith the. characteristics of the desired arbitrary function to begenerated, so that upon operation, the output function of the rotorshaft angle is either, (1) an electrical signal representing a synthesisof the desired arbitrary function, or (2) a restraining or opposingtorque on the output shaft varying in accordance with said arbitraryfunction.

The basic theory underlying the present invention for attaining theseand other objects and advantages, stems from the fact that any arbitrarymathematical function that is continuous over a finite period may beexpanded into a Fourier series of sines and cosines. Sinceelectromagnetic resolvers, such as described above and in the foregoingmentioned patent, are already known in the art and are individuallycapable of generating sinev and cosine functions, it will. be realizedatonce that a series of such resolvers individually designed to generatea sine or cosine of given amplitude and shaft angle frequency, may havetheir outputs combined together to provide a function that is the.summation of such. sines and cosines- BY changing the amplitude andspace frequency of each indi- 2,872,602 Patented Feb. 3, 1959' 2. vidualgenerator in accordance with the separate terms in the Fourier expansionof an arbitrary function, it is possible. to provide acombined.or'summed output which represents the arbitrary function itself.

In accordance with the present invention, a single electromagnetictransfer device. comprising a stator and rotor is provided which willimmediately yield the. desired arbitrary function. This result isachieved by distributing the number of conductor. windings in. thestator and the rotor slots in a manner determined by the coefficients ofthe sine and cosine terms in the Fourier series expansion of the.arbitrary function. It ispossible, accordingly, to provide in a. singledevice, an output signal or shaft torque that represents any desiredarbitrary function of shaft angle, including the elementary sine. andcosine functions.

A better understanding. of this; invention will be had by referring tothe accompanying. drawings, in which:

Figure 1 is a schematic cut-away perspective view of an electromagnetic;transfer device showing stator and rotor bodies coaxially positioned;

Figure 2 is an enlarged cross-section of a portion of the stator and;rotor bodies of an. electromagnetic transfer device, such asschematically shown in Figure. 1;

Figure 3 is a schematic plan view of the. stator illustrating theconductor distribution in accordance with an illustrative example of theinvention; and,

Figure 4 is a similar, schematic plan view of the rotor for the exampleillustrated in Figure 3.

Referring to Figure 1, there is shown shaped stator body 10 having acentral opening within which is coaxially positioned a rotor body 11.The rotor 11 is secured to a rotor shaft 12 and is adapted to be rotatedto assume different. angular positions 7. with respect to the stator. Asshown, the stator is provided with aplurality of radially cut statorslots 13, and the rotor is similarly provided with a plurality ofradially cut rotor slots 14. The sets of slots 13 and 14 are adapted tore ceive conductor windings. whereby the stator body may establish anelectromagnetic field across the air gap 6 to which the conductorwindings of the rotor body are inductively coupled, or vice versa.

In Figure 2, there are shown symbols. c c

3. 4 designating the per unit. conductor density in the. respectivestator slots 13, and symbols d d designating the per unit conductordensity in the respective. rotor slots 14. Let c equal the per unitconductor density in the s stator slot, and ai equal the per unitconductor density in the r rotor slot. The actual number of con;-ductors in the s and r slots is. determined by multiplying the absolutevalues of' 0 and d,. by the total number of conductors employed in thestator and the rotor respectively.

Stated another way, the per unit conductor density, e in the s statorslot, is, in magnitude, the ratio of the actual number of physicalconductors in the s stator slot to the, total number of net effectiveconductors in all the stator slots, comprising the particular statordistributed winding made up of a group of c with a plus sign. prefixedfor one chosen (reference) direction of actual current flow in theconductors in the slots, and with a minus sign. prefixed for theopposite direction of actual current flow in the. conductors in slotshaving. Such cur rent flow opposite to the reference direction. The samesituation obtains for the a, per unit conductor density in the r rotorslot. The factor times the per unit, conductor density is the percent,conductor density in a particular slot.

In accordance with the present invention, the conductor windingdistribution among the slots of the stator and rotor; that is, the:actual values of the 0 and d l are uniquely predetermined bythecoefficients of the Fourier Series expansion of the arbitrary functionto be generated.

a cylindrically' With such distribution, the variation of the ratio ofthe output signal to the input signal or the variation of the rotorshaft torque with changes in the rotor shaft angle may be made torepresent the desired arbitrary function. .Themanner in which and d areuniquely related to the coefiicients of the Fourier series in order todetermine the distribution of the conductors in the slots will now bedescribed.

Referring to Figure 2, let the dashed line 15 represent a. referenceangle line for the stator 10. Any point in the airgap 6 having a certainfiux density generated by the stator windings alone may then be locatedby specifying an angle 0 so many degrees from the reference line 15.

Similarly, let the dashed line 16 represent a reference angle line forthe rotor. Any point in the airgap 6 having a certain fiux densitygenerated by the rotor windings alone, maythen be located by specifyingan angle 1; so many degrees from the reference line 16.

The angle of shaft rotation 7 representing the degree of rotation of therotor with respect to the stator, will then be the angle between thereference lines 15 and 16 as shown in Figure 2.

.The arbitrary function to be generated in accordance with the inventionwill be represented by changes in the output of the device withvariations of therefore, let f('y) represent the desired arbitraryfunction. v

In the case where it is desired to synthesize the arbitrary function asan electrical output signal,

no g n) (1) where E =output voltage from the device, and E =inputvoltage to the device.

In the case where it is desired to provide a shaft torque which variesin accordance with an arbitrary function, this torque will beproportional to the product of applied stator and rotor voltage, as wellas a function of the shaft angle, or:

where T(-y) represents the arbitrary torque function, =voltage appliedto the rotor, and 1 Eg =voltage applied to the stator.

f(v)=2 cos w where n=harmonic order, and A are the Fourier coefficients.For these cases:

Let f(0) equal the normalized stator distribution of flux densitygenerated in the airgap 6 for different angles of position '6 around thestator periphery, referred to the stator. reference line 15, when thestator conductor distribution alone is excited.

Let f(1;) equal the normalized rotor distribution of flux densitygenerated in the airgap 5 for different angles of position 1 around therotor periphery, referred to the rotor reference line 16, when the rotorconductor distribution ,alone is excited.

f(0) and f(1 may then be represented by Fouriers Series as follows:

f(n)= /2E n i y where na nr equal the n harmonic amplitude for thestator and rotor 1 respectively.

From Figure 2 it will he noted that:

Now, the convolution integral of f(0) and f(1;) for :09-1 is given, ingeneral, by

I fine-no Upon the substitution of 0- for 1;, from Equation 6, and f(0)and fi as given in Equations 4 and 5, into this convolution integral andintegrating, there results:

' nr nr A E T 005 TL/ nn A. .(s)

.whereby the arbitrary function, f('y)- of Equation 3 is given by Thenext step is to relate the quantities A and A, to theper unit conductordensity in all stator and rotor slots as defined generally by C and dfor the s and r slots respectively;

Referring again to Figure 2, assume the stator has a total of p slotsdistributed over 21r radians in 0 and that the rotor has a total of qslots distributed over 21r radians in 1;.v The per unit slot-conductordensities c c c c c o where c represents the per unit slot-conductordensity in the last slot of the p stator slots, are distributed in thesep slots respectively, adjacent slots being separated 21r/p radians.Similarly, the per unit slot-conductor densities d d d d where drepresents the per unit slot-conductor density in the last of the "qrotor slots, are distributed in these q slots, respectively, adjacentslots being separated by 21r/q radians.

In the case of continuous windings on both the stator and rotor, twoconditions must be met:

and,

where each c and d is given a positive or negative sign.

Additional constraints further reducing the number of a e-anus a is, aknown constant for a given, 1) and n and for known axes and locations ofaxes of symmetry and asymmetry of the stator slot-conductor densitydistributed array, and varies at most, only withn;

3,, is a known constant for a given q and n and for known axes andlocations of axes of symmetry and asymmetry of the rotor slot-conductordensity distributed array,

Y and varies at most, only with n;

C is, as defined previously, the per unit stator slot-conductor density,with algebraic sign, in the s stator slot;

d is, as defined previously, the. per unit rotor slot-conductor density,with algebraic sign, in the. r rotor slo g (s,n,p-) denotes the statorgenerating function whose form is known and determined by the knownaxesand locations of axes of. symmetry and asymmetry of the statorslot-conductor distributed ordered array over 21r radians. in 0; andwhichis a function of the slot number s on the stator, of the order n ofthe space harmonic referred to Zn radians in 0 as the fundamental orfirst harmonic, and of p, the number of equispaced slots in the stator;and,

g (r,n,q denotes the rotor generating function whose form. is known anddetermined by the known axes and locations of axes. of symmetry andasymmetry of the rotor slot-conductor distributed ordered array over 27!radians-in 1 and which is. a function of the slot number r, on the.rotor, of the order n of the space harmonic referred to 211 radians in'1 as the fundamental or first harmonic, and of q, the number ofequi-spaced slots 1 in the rotor.

For known 2 and q" and for known axes and locations of axes of symmetryand asymmetry of the stator and rotor slot-conductor density distributedarrays, Equations 12 and 13 yield a set of simultaneous equations asfollows:-

order, n: of harmonic amplitudes; A to approximate; the

given function to within prescribedtolerances over a prescribed range ofthe independent variable 7. These A s then, are given knowns in the setof simultaneous Equations 14. Further, the a s, fl s, g s and g s areknown. Only the c s and d s are unknowns in the set of Equations 14 andtherefore these equations may be simultaneously solved to provide thevalues of the c s and d,.s, there being as many equations as there areun- 4 knowns.

In other words, if the number of the highest harmonic amplitude, A,,(not the order), is denoted by N, and the least number of independent cs and dgs to synthesize the arbitrary function required, are denoted byS and R respectively, then:

and therefore, it is possible to solve for each c and each d d fromEquations 14.

Conductor windings are then applied to the stator and rotor slots 13 and14 in accordance with the solutions for c, and d,. When an input signalis then applied to the device-to excite the particular distributedwinding array, the electrical output signal will vary with the shaftangle in accordance with the. initially selected arbitrary functiOnf('y).

It will be recalled that the above analysis is for generating anarbitrary function which may be expressed by a Fourier series ofcosines, Equation 3. Another class. of arbitrary functions may berepresented by a Fourier series of sines, thus:

The analysis for relating c and d to the Fourier coeificients, B,,, inEquation 16, is carried out in an identical manner. to that given abovefor Equation 3, except. for a rotation of 1r/2: radians of the 0 andreferencelines.

in the most general case in which the arbitrary function is representedby both summations of the Fourier series, thus:

f('y) =2 (A,, cos 11 3,, sin 11 (17) the c s and d s determined by the Bs, may be superimposed on the c s and d s determined by the A s, re-

sulting in a single, composite stator sl'otconductor densitydistribution on the stator, and ina single, composite rotorslot-conductor density distribution on the rotor.

It will thus be seen that the conductor windings in the different slotsare provided with weighted operating characteristics from the standpointof the number of conductor windings in each slot and that such weightedoperating characteristics are in accordance with the solution of aplurality of simultaneous equations. Each simultaneous equation involvesthe weighted operating characteristics of a different harmonic generatedby the windings in a pair of Fourier Series. The product of the pair ofFourier Series expresses the arbitrary function to be generated withprogressive displacements between the. rotor. and, the. stator.

Furthermore, each Fourier Series in the pair individ- Considering nowthe case of a torque transfer arbitrary :function generator as describedby Equation 2, the restraining or opposing torque function is given by:

.where L ('y) ideally denotes the coefficient of mutual inductionbetween the stator and rotor as a function lof the shaft angle 7, and Iand I respectively are the current flowing in the stator and rotorwindings as a result of the "applied voltages E 'and E, of Equation 2.

Assume first, that the arbitrary torque function is expressible by aFourier Series of sines, thus:

whereLA' denote the Fourier coeifi cients of the torque function. J

With respect to the A and A of Equations 4 and 5 of the voltage transferdevice, set

where K is the proportionality constant and is independent of 'y, I,,,and 1,. The arbitrary torque function of y given by Equation 18, is thengiven by:

I Ten-K U m) Fon in which again, 'y=0-1 resulting in:

As in the case of the voltage transfer arbitrary function generatoranalysis, Equations 19 and 26 can be equated term by. term, by making:

I I L-171 By dropping the primes in Equation 27, it is evident that avoltage transfer device whose normalized harmonic amplitudes are givenby:

is also a torque transfer device whose normalized harmonic amplitudesare given by ns' nr and whose reference lines for 'y in the outputfunction are space phase displaced 1r/2 radians with respect to thereference lines of the voltage transfer device (lines 15 and 16 inFigure'2).--

To determine c and d, for the torque transfer device, a set ofsimultaneous equations whose left hand sides are identical to those inthe set of Equations 14, but whose right hand sides in the general term,are written nA instead of n A are used. Thus, the arbitrary torquefunctions may be reproduced by distributing the stator and rotorconductor windings in accordance with the new c s and d s.

The analysis for relating c and d to the Fourier coefiicients for atorque function expressible by:

- T(y)=ZB,', cos m 28) is executed in an identical manner, except for arotation of 1r/ 2 radians in the 0 and 7 reference lines.

In the most general case for' the torque function,

wherein, I

T( )=E(A,', sin n-H-BI, cos n'y) (29) the c s and dfs determined by theB s, may be superimposed on the c s and d s determined by the A' s, asin the case of the voltage transfer device, resulting in a singlecomposite c stator slot-conductor density distribution on the stator andin a single composite d, rotor slot-conductor distribution on the rotor.

Summarizing, the preceding formulas elicit the unique stator and rotorper unit slot conductor density distributions over all slots for thegeneration of the given arbitrary empirical or analytical function asavoltage or torque transfer function within the limitations of the totalnumbers of stator and rotor slots. Thus, having once determined thestator and rotor per unit slot conductor densities, they may berespectively multiplied by the total number of net efiective statorconductors and the total number of net effective rotor conductors toyield the actual number of conductors in each slot of the stator androtor and the relative directions of current fiow therein.

The total number of actual conductors and the wire size employed on anystator or rotor winding in slots of a given area is determined, amongother factors, by the total slot volume, the slot fill factor,insulation thickness, ampere turns for proper non-saturating use of themagnetic material, air gap width, maximum number of H conductors in themaximum filled slot, transformation for use in connection with functionshaving sine syme metry about the origin, :0. In the example underconsideration, assume the arbitrary function to be generated isexpressed by:

This function is very useful in modern computers and guided missilecontrol systems and is an example of the latter above mentioned type offunction having sine sym- Assume that it is desired to develop thisfunction in a -0.636620 f(7) +0.636620 whereby one servomechanism andone resolver or potentiometer in a conventional computer network may beeliminated. V The corresponding range in "y is In order to develop the Bharmonic amplitudes of the given functionwhich step is a prerequisiteutothe unique i v fi fin The usual Fourier analysis yields the followingharmonic amplitudes through the 35th order, 11:35. Since the functionhas sine symme ry, only the odd. harmonics in the Fourier expansion needbe computed:

10 possible as limited only by the number of stator and rotor slotsavailable and the physical constraints of electromagnetic inductionfields.

Since it has been initially recognized and noted that the function to besynthesized has sine symmetry about the origin, :0, the stator and rotorstep-wise flux density distributions (mmf functions) are chosen fromthat class which also has this same sine symmetry but not sinusoidalshape. Thus, letting p equal the number of stator slots and q equal thenumber of rotor slots, the general Equations 12 and 13 for this class offunctions, respectively take the forms:

The K factor in Equation 31 is the so-called per unit skewness andconstitutes that fraction of one slot-to-slot width that the rotor stackskews in going from the front end of the rotor body to the rear end ofthe body; a is defined by c,=a- -a h(s) is a distribution function overone quadrant of the stator slots which is repeated over the other threequadrants; m(r) is another distribution function over one quadrant ofthe rotor slots which is repeated over the other three quadrants; s, asbefore, is the sequential numbering of the stator slots in one quadrant;and r is the sequential numbering of the rotor slots in one, quadrant.Specifically, for the case of 1:20 stator slots and q: 12 rotor slots,the above equations become explicitly:

The general Equations 14 for sine symmetry are then expressed by B -B =nB, and may be Written out:

sin 5:5 T r=d (28'l)'rr H 12 i =1 a sin 20 KW '21 d, 11.1 6 -B sin K 1(28,1)31r k 1 7:3 T'n' 8 1 a. sin 20 T 1 d, $111 -9B3 Sin 3 (281)51r 12T511 8 1 a, sin 20 7 1. d, sin 25B;

- Be cause the c s (as expressed in terms of a and d s .are normalizedto unity:

l tl= ,E .=1 (35) v s=1 and, V v

r=3 -Ed,.=l (36) a; 0.102847 a -0.424190 0, a,a., 11-; 0.251114 I a2.198569 K =0.890003 c; 0.192847 -0.617037 03 0.765304 0 1.947455 c3.l98569 Plu 1.000000 In general, for this case of 1:20 stator slots andq=l2 rotor slots, the generated B obtained by substituting the above K,a! and c values back in to Equations 34 will be exactly equal to theoriginal 13 calculated previously for the given function for 11:1 to 35only over the order of n from 1 to inclusive utilized to obtain the K,d, and 0 values. However, they are the unique rotor skewness and rotorand stator conductor distributions for the generation of the givenfunction,

1-cos 7 'Y over the specified range, for the case of 12:20 stator slotsand q=12 rotor slots with maximum harmonic matching for this slotcombination.

A comparison between the calculated B of the given function and thegenerated B of the function-generator is set forth below:

11 B11 Calculated B Generated This table estahlishes'the validity of thestatement that the harmonics, of the original Fourier Series of thefunction and the harmonics of the generated function match exactly, andexactly only, up to and through n=15, the maximum possible and uniquematching within the limits of the slot number's, 17:20, q=l2. V

It is desirable to finally compare the normalized function valuesdeveloped by the function generator, with the normalized values of thegiven function. This is done, utilizing the first 35 harmonics, thoseabove that having dropped to very small values and not contributingsubstantially to the error percentage.

i degrees fin fin generated Percent.

, rror.

Nowhere except in the last interval does the percent error exceed 0.5percent for this rather unusual heretofore .unavailable, practicalfunction, and the R. M. S. error is less than 0.08%. Since even thisremaining error is quite periodic, it may be still further minimized bya simple auxiliary winding designed on the above principles for the mainfunction' The configuration for the rotor per-unit slot conductordensities demonstrates a ratio of 4.451504 between the maximum loadedslot (r=1) and the least loaded slot (r=3 On the stator, the situationis somewhat difiercut, because some of the slots in a quadrant haveconductors which are oppositely poled (reversed current direction) toothers in the same quadrant of slots. Since summation of the absolutemagnitudes of C is the base reference for the total number of physicalconductors in one quadrant of the stator slots, and equals 6.631212,whereas the absolute magnitude of the summation of 0 is always equal tounity, this winding requires a total number of physical conductors equalto 6.631212 times the total number of net effective conductors. Theratio between the maximum loaded slot (s=5) and the least loaded Slot(s=1) is 16.586044.

are in the maximum loaded slot, s=5. Postulating 60% fill factor,including slot insulation, wire insulation, lay

factor, and so forth, the copper area of 400 conductors in the slot s=5equals 0.014130 square inch. Hence, a wire size for which the copperdiameter is 0.0126 inch, is specified, for example, AWG No. 28.

Referring now to Figure 3, the physical stator conductor distribution inthe stator slots is shown by the numerals which indicate the actualphysical number of conductors in each slot. The dots indicate that thedirection of current flow in certain slots is coming out of the drawing,while the crosses indicate that the direction of current flow in certainslots is passing into the drawing. The, actual windings may be toroidaLlatitude, or effected in any other convenient manner. The step-wiseair-gap flux density distribution (mmf. function) generated by thisWinding over 360 of the inside periphery of the stator, directlycorresponds to the step-wise poled conductor distribution given inFigure 3.

Calculation of the required self-inductance of the rotor winding in apractical size 23 case, for an air-gap 6 of .00554 inch and for anexisting rotor slot area of .02461 square inch necessitates a total noteffective number of conductors per rotor Winding of 2000. This numbercorresponds to the actual total number of physical conductors in onequadrant, since the direction of current how in any one quadrant isuniform. Thus, there are 500 conductors per rotor quadrant.

The corresponding poled conductor distribution is illustrated in Figure4 wherein a 60% fill factor is used and No. 28 AWG copper wire isemployed. As in the case of Figure 3, the numerals indicate the actualphysical number of conductors in each slot and the dots and crossesindicate the corresponding direction of current flow.

The above specific example thus illustrates the manner in which the slotconductor density distributions for the stator and rotor are determinedin accordance with the characteristics of the arbitrary function wherebythe output of the generator represents the arbitrary function.

It will be seen from the foregoing description that the presentinvention provides an electromagnetic transfer device which is far more.versatile than any heretofore known in the art, in that it may beemployed to generate any arbitrary continuous function, reproducing suchfunction as either an electrical voltage signal or as a shaft torque.

What is claimed is:

1. An arbitrary function generator comprising: a first body; a secondbody co-axially positioned in concentric relationship to said first bodyfor rotation relative to said first body through a variable angle 7measured from a radial: reference line on said first body, said secondbody and first body defining an annular air gap therebetween; said firstbody having a plurality of discrete radially cut slotsciitcumferentially spaced about its axis; electrical conductorspositioned in said slots; each slot, in sequential order from saidreference. line containing a discrete number of said electricalconductors such that the ratio of the number of conductors in any oneslot to the total number of conductors in all of said slots defines aunique absolute per unit conductor density in said any one slot; meansfor passing current through said conductors in specified directions toestablish a first body slot step-wise flux density distribution in theadjacent air-gap varying over different circumferential points measuredfrom said reference line; and conductor means on said second bodyadapted to be inductively coupled to said first body step-wise slot fiuxdensity distribution upon rotation of said second body through variousvalues of the angle y, the effective per unit i slot conductor densitiesin said first body slots as derived from the absolute per unit conductordensities and said specified directions of current flow, having values,such asto: provide said step-wise flux density distribution withharmonic amplitudes and signs equal to the term by term co-efiicients ofthe Fourier Series expansion of said arbitrary function, whereby thesignal induced in said conductor means of said second body represents afunction of body for rotation relative to said stator body through avariable angle 7 measured from a radial. reference line.

on said. stator, said rotor and stator defining'an annular air-gaptherebetween; said rotor and stator bodies each having a plurality ofdiscrete radially cut slots circumferentially spaced about theirrespective axes; electrical conductors positioned in said rotor andstator slots; each slot in said rotor and stator containing a discretenumber of said electrical conductors such that the ratio of the numberof conductors in any one rotor slot to the total number of conductors inall of said rotor slots defines the absolute per unit conductor densityin said any one rotor slot and the ratio of the number of conductors inany one stator slot to the total number of conductors in all of said Q 0d S th absolute per unit conductor density in said any one. stator slot;and means for passing current through said conductors in each of saidrotor and stator slots in specified directions to establish a rotor andstator slot step-wise flux density distribution in the adjacent airgapvarying over different circumferential points measured from. saidreference line; the respective electrical conductors associated, withsaid rotor and stator being inductively coupled upon rotation of saidrotor through various values of the angle 7, the products of a firstfunction of the effective per unit conductor density distribution ineach of said rotor slots with a second function of the eifective perunit conductor density distribution in each of said stator slots, asderived from said rotor and stator absolute per unit conductor densitiesand said rotor and stator slot specified directions of current flow,being term by term respectively proportional to and of the same sign asthe term by term coeificients of the Fourier Series expansion of saidarbitrary function, whereby the output signal of said generatorrepresents a function of said angle 7 corresponding to said arbitraryfunction.

3. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of av pair of transverseaxes, a first member, a second member movable v relative to the firstmember, a first plurality of generating means disposed at spacedpositions along the first member and a second plurality of generatingmeans disposed at spaced positions along the second member, thegenerating means in the first plurality being interrelated and beingprovided with, characteristics. to generate, upon progressivedisplacements between the first and second members, the desiredarbitrary function as indicated by a convolution product integral whichrepresents the integral of the product of a pair of functions, onerepresenting the characteristics of the signals generated by the firstgenerating means upon progressive relative orientations of the firstmember and the other representing the characteristics of the signalsgenerated by the second generating means upon progressive relativeorientations of the second memher.

4. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member provided with a plurality of apertures at spacedpositions. along one periphery, a second member provided with aplurality of apertures at spaced positions along the; periphery facingthe apertures in the first member, a first plurality of generating meansdisposed in co-operative relationship with the apertures in the firstmember and av second plurality of generating means disposed inco-operative relationship with the apertures in the second member, thegenerating means in the first plurality and the generating means in thesecond plurality being interrelated and being provided with weightedoperating characteristics to produce in the different apertures fielddensities dependent upon the polarities and magnitudes of thecoefiicients in a pair of Fourier Series which together represent thearbitrary function to be generated upon progressive displacementsbetween the first and second. members and which individually representarbi 15 trary functions to be generated upon progressive relativeorientations of the first and second members.

5. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member,

a second member movable relative to the first member, a first pluralityof generating means disposed at spaced positions along the first memberand a second plurality of generating means disposed at spaced positionsalong the second member, the generating means in the first plurality andthe generating means in the second plurality being interrelated andbeing provided with weighted operating characteristics in accordancewith the solution of a plurality of simultaneous equations eachinvolving the weighted operating characteristics of a different harmonicgenerated by the generating means in the first and second pluralities inrelationship to the product of the coefiicients of the particularharmonics. in a pair of Fourier Series the product of which expressesthe arbitrary function to be generated with progressive displacementsbetween the first and second members and which individually expressarbitrary functions to be generated with progressive relativeorientations along the first member or along the second member.

6. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member made from magnetic material and provided with aplurality of slots at spaced intervals along one periphery, a sec ondmember made from magnetic material and 'disposed for movement relativeto the first member and provided with a plurality of slots at spacedintervals along a periphery facing the first member, and a firstplurality of windings disposed in the different slots in the firstmember and a second plurality of windings disposed in the differentslots in the second member and provided with conductor densities andwound with polarities in accordance with the solution of a plurality ofsimultaneous equations each involving the relationship between theconductor densities contributed by the different terms toward aparticular harmonic and the coefficient of that harmonic in a FourierSeries which represents the arbitrary function to be generated uponprogressive displacements between the first and second members.

7. In apparatus for generating any'desired arbitrary function havingonly a single value at each position along along one of a pair oftransverse axes, a first member made from magnetic material and providedwith a plurality of slots at spaced positions along one of itsperipheries, a second member made from magnetic material and disposedfor'movement along the first member and provided with a plurality ofslots at Spaced positions along a periphery facing the slots in thefirst member, the number of slots in the second member being ditferentfrom the number of slots in the first member for corresponding units ofdistance, and a first plurality of windings disposed within thedifierentslots in the first member and a second plurality of windingsdisposed within the difierent slots in the second member, the windingsin the first plurality being disposed and interconnected and thewindings in the second plurality being disposed and interconnected toprovide in the different slots conductor densities representing thesolutions of a pair of Fourier Series the product of which representsthe arbitrary function to be generated upon progressive displacementsbetween the first and second members and which individually representthe functions to be generated by the windings in the first plurality andthe windings in the second plurality upon such progressivedisplacements.

8. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member made from magnetic material and provided with firstand sec- 1% ond peripheries and provided with a plurality of slots atspaced intervals along one periphery and provided with a continuousportion at its other periphery for the flow of magnetic flux, a secondmember made from mag netic material and provided with first and secondperipheries with the first periphery facing the slots in the firstmember and provided with a plurality of slots at spaced intervals alongthe first periphery and with a continuous portion at its other peripheryfor the flow of magnetic flux, and a first plurality of windingsdisposed in the different slots in the first member and a secondplurality of windings disposed in the different slots in the second'member and provided with conductor densities and wound with polaritiesin accordance with the convolution product integral of a pair offunctions which individually represent the relationship to be generatedfor progressive orientations along the first and second members and theintegrated product of which represents the arbitrary function to begenerated for progressive displacements between the first and secondmembers.

9. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member made from magnetic material and provided with ahollow annularconfiguration and with a plurality of slots at spacedpositions along one of its annular peripheries, a second member madefrom magnetic material and provided with a hollow annular configurationand disposed in concentric relationship to the first member for movementalong the first member and provided with a plurality of slots at spacedpositions along the annular periphery facing the slots in the firstmember, the number of slots in the second member being different fromthe number of slots in the first member, and a first plurality ofwindings disposed within the different slots in the first member and asecond plurality of windings disposed within the difierent slots in thesecond member, the windings in the first plurality being disposed andinterconnected and the windings in the second plurality being disposedand interconnected to provide in the different slots conductor densitiesrepresenting the convolution product integral of a pair of functionseach of which represents the particular relationship generated forprogressive orientations of a different one of the first and secondmembers and the integrated product of which represents the desiredarbitrary function to be generated for progressive displacements betweenthe first and second members.

10. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first hollow annular member made from magnetically receptivematerial and provided with slots disposed at spaced positions along theinner periphery of the member and with bars between the slots, a secondhollow annular member made from magnetically receptive material andprovided with slots disposed at spaced positions along one annularperiphery and with bars between the slots, a first plurality of windingsdisposed in the difierent slots in the first member and connected in acontinuous circuit with the other windings in the plurality, and asecond plurality of windings disposed in the different slots in thesecond member and connected in a continuous circuit with the otherwindings in the plurality, the first and second pluralities of windingsbeing constructed to provide conductor densities in the different slotsin the first and second members for the generation of the desiredarbitrary function upon progressive displacements between the first andsecond members and in accordance with the solution of a plurality ofsimultaneous equations involving the conductor densities in theclifierent slots and the coeflicients of the successive terms in a firstFourier series expressing a particular function to be generated withprogressive displacements along the first member and the coefiicients ofthe successive terms in a second Fourier series expressing a particularfunction to be generated with pro gressive displacements along thesecond member wherein the first and second Fourier series are providedwith interrelated characteristics to have their product represent thedesired arbitrary function.

11. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member made from magnetically receptive material andprovided with an annular configuration having an opening at the centerand provided with a plurality of slots spaced annularly around theperiphery of the first member, a second member made from magneticallyreceptive material and provided with an annular configuration having anopening at the center and disposed in concentric relationship to thefirst member for rotation relative to the first member and provided witha plurality of slots spaced annularly around the periphery of the secondmember at the periphery facing the first member, and a first pluralityof windings disposed in the slots in the first member and a secondplurality of windings disposed in the slots in the second member, thewindings being connected and being disposed in the different slots inparticular patterns to produce in the different slots conductordensities dependent upon the polarities and magnitudes of thecoefiicients in a Fourier Series representing the arbitrary function tobe generated upon progressive displacements between the first and secondmembers, the windings in the first plurality being connected anddisposed to produce a resultant value of zero for all of the conductordensities in the diflerent slots in the first member and the windings inthe second plurality being connected and disposed to produce a resultantvalue of zero for all of the conductor densities in the difierent slotsin the second member.

12. In apparatus for generating any desired arbitrary function havingonly a single value at each position along one of a pair of transverseaxes, a first member made from magnetic material and provided with ahollow annular configuration and with a plurality of slots at spacedpositions along one of the annular peripheries, a second member madefrom magnetic material and provided with a hollow annular configurationand disposed in concentric relationship to the first member for rotationrelative to the first member and provided with a plurality of slots atspaced positions along the annular periphery facing the slots in thefirst member, the number of slots in the second member being differentfrom the number of slots in the first member, a first plurality ofwindings disposed in the different slots in the first member and asecond plurality of windings disposed in the difierent slots in thesecond member, the windings in the first and second pluralities beingconstructed to provide the slots with conductor densities dependent uponthe magnitudes and polarities of the coefiicients in successive terms ofa pair of Fourier Series which individually represent progressiverelative orientations along the first and second members and whichtogether represent the arbitrary function to be generated forprogressive displacements between the first and second members.

References Cited in the file of this patent UNITED STATES PATENTS618,578 Newcomb Jan. 31, 1899 1,308,041 Chubb July 1, 1919 1,807,218Langewiesche May 26, 1931 2,488,771 Glass Nov. 22, 1949 2,590,845 CurryApr. 1, 1952 2,719,930 Lehde Oct. 4, 1955 2,731,574 Soredal Jan. 17,1956 OTHER REFERENCES Book: Alternating Current Circuits, by Kerchnerand Corcoran, 3rd edition; John Wiley & Sons, New York, 1951, pp.163-188.

